- Email: [email protected]
On the precision of momentum measurements with a magnetic spectrometer
NUCLEAR INSTRUMENTS AND METHODS 97 (I97I) 315--318; © NORTH-HOLLAND PUBLISHING CO.
ON THE PRECISION OF MOMENTUM
MEASUREMENTS WITH A MAGNETIC SPECTROMETER A. FROHLICH
Department o f Physics and Astronomy, TeI-Aviv University, R a m a t Aviv, Tel-Aviv, Israel
Received 11 June 1971 Methods of analysing the data from magnetic spectrometers are examined. The effects of Coulomb scatter counter efficiencies and assumed spectrum exponent on the observed spectra are
considered, and how they might be incorporated into the analytic methods used for correcting the measured muon flux and charge ratio.
1. Introduction
scattering within the magnet iron and that due to the efficiency and spatial resolution o f the particle detectors. In this paper the m o m e n t a are considered to be high enough that the m o m e n t u m loss in the iron is negligible. When the thickness of the iron is much greater than the radiation length, as is usual, the projected multiple scattering o f relativistic muons has a Gaussian distribution with a standard deviation o f 6)
The study of high energy cosmic ray muons is, at present, the only means o f investigating m u o n production models and interactions at energies o f the order o f magnitude of 1 TeV (1012 eV). The principal instruments used in these experiments are magnetic spectrometers, the majority incorporating solid iron magnets1'2). However the accuracy o f measurements with these instruments has not been fully investigated. A quantity called the m a x i m u m detectable m o m e n t u m , mdm, is variously defined and used as a measure o f the upper limit o f the m o m e n t u m resolution, although measurements at higher m o m e n t a have been made and compensated for instrumental errors3). These corrections have been criticised as being insufficient to ensure enough accuracy~). It is precisely where these corrections become important, at m o m e n t a higher than or o f the same order as the mdm, that the collection statistics are p o o r because of the steep decrease of the m u o n flux with m o m e n t u m , and also where effects such as m u o n isotropy m a y appearS). This work was initiated as part o f the design studies for a large m u o n spectrometer that is to be built at "Fel-Aviv, in order to estimate the precision attainable.
11.287 x 10 - 6 a~ =
/
. ,,' x
rad.
(3)
P The spatial resolution o f a spark chamber also has a Gaussian distribution with a standard deviation of 7) o-c. The efficiency o f operation o f these chambers depends on the operating conditions and the number o f tracks. Fig. 1 shows a schematic outline of a simple
V////////////J//////////////////////////////////////////~//J~
2. Sources of error
The experiments consist o f measuring the angular deviation o f a muon within the magnetic volume and using the equations p = k/0,
(1)
k = 3 x 10 -~° B . x .
(2)
,where B is the mean flux in gauss, x the length o f the trajectory within the magnet and p is in TeV/c. The measurement o f the angular deviation 0 is subject to two kinds o f error; that due to C o u l o m b 315
t
I3
Fig. 1. Schematic spectrometer.
316
A. FROHL1CH
spectrometer with four trays of detectors, each tray containing n spark chambers relatively close together. If the efficiency of each gap is 100% the scatter of the measured angular deviations will be Gaussian with a standard deviation a m = 2ad(Dx/n) rad, without scatter, and the overall standard deviation of the measurement will be a = (a~ + am2){ rad.
(4)
In practice the efficiency of spark chambers is less than 100%. A gap can either produce an acceptable spark(s) or not, that is the probability distribution is binomial. As an example the probability that m out of 12 gaps will fire is
12Cm(~)rn(1--q)12-m with ~ the gap efficiency. With a gap efficiency of 98% all twelve gaps will fire together only in about 80% of the events. Consequently the measurement error a., should be weighted accordingly, to all of the possible combinations of acceptable firings. The probability distribution of the angular deviations resulting from a muon of momentum p = + k/O is P(~b) dq~
/3 is often referred to as the maximum detectable momentum, mdm, the Durham group preferring the value 1.48/3. This phrase is rather misleading because meaningful measurements of momenta greater than this value are feasible. It is suggested that an expression of the type "equivalent noise m o m e n t u m " would be semantically better. 3. Corrections to the flux spectrum
Because of the above effects corrections have to be made to the final observed spectrum. It is assumed that the most probable incident flux is N(p)dp=FP-~dp with a constant value of Y in the range Pn,~,, ~ P ~< oo, and that the value of k in eq. (2) is also constant within this momentum range. Using eq. (5) the observed deflection spectrum would be N'(~b) .
. . . exp - ½ Jo x/(2n)a
+
c
,](2n) a
3.5
If the path lengths of measured tracks within the magnetic volume do not vary appreciably, i.e. k in eq. (2) is constant, one can transform this distribution from the angular plane to the momentum plane:
P(p) d p -
\/2n--
/
exp 2.5
where/3 = k/a and fi = k/O. As can be seen this distribution is not Gaussian; the most probable momentum satisfies the equation
?'=3
2.8
2
2+8
,
(P) = 4
assuming scattering to be negligible. This constancy of k is only valid at high momenta and where the aspect ratio of the spectrometer is large. With a large acceptance aperture the variation of k due to variations of the path length x should be taken into account: not only here but also in the final analysis when converting angles to momentum with the aid of eq. (!). For this reason working in the angular plane is preferred.
2.6
1.5 2.4
2.2 1
7=2 1
2
3
P@
Fig. 2. Correction factor to the flux spectrum with scattering negligible.
ON THE PRECISION
OF M O M E N T U M
using a mean value of a. The second term results from the probability of an apparent charge reversal. The correction factor is then C(qS)= N'((9)/F'W-2 within this range of 0. The value of 0m,x is the limit where one or more of the above assumptions are no longer valid; such as an appreciable variation of k, or where scattering effects predominate such that cr varies with 0, or where the magnetic cut-off of the instrument is no longer negligible. Eq. (7) can be easily solved analytically when y equals 2 or 3: For y = 3 C = e r f ( ~b ~ +½{erf(tl)-erf(t2)} \x/2 ~1 +
a
1
d?x/(2n) 2expI-( /-~-~)21- exp[-(tl)2] -
- exp [ - (t2)2],
(8)
when tl = ,,,/21
, and t2
x / 2 \ tr +
"
For 7 = 2 C = ½[erf(tl) + err(t2)]. Hayman and Wolfendale 3) have used eq. (8) with 0ma x = OO. Even if one can ignore the end effects denoted by a finite 0m~x the actual values of C are sensitive to the assumed value of ),. Fig. 2 shows C as a function of p//3 and 7 under these conditions. One would expect a finite 0max when scattering effects become appreciable, that is when as> am o r Pmin
~ 5.64 x
Using for the dimensions of fig.l x = 3 0 0 c m , D = 250 cm, n = 3 and ac =0.5 mm and assuming a spark efficiency of 100% we find pmi,<0.8 TeV/c. In this situation, where cr~trs, using eqs. (1) and (3) we arrive atp/p = a/q5 = (3.76 x 104)/(Bw/x),a constant for a given trajectory. Thus for measurements at low momenta, with a constant path length in the magnetic flux the limits of eq. (8) are t-
532
effects, to their measurements by the )(2 test. This method will, of course, only optimise the measurements to within the parameters of the flux model chosen a priori. This criticism applies to all of the techniques chosen to correct spectra. If, for example, the exponent changes, as the results of the Utah group would indicateS), these corrective techniques, having a fixed 7, might not show this: instead they would give a best fit to the data from a model having a fixed exponent. Any anticipated changes in y would have to be allowed for by using a finite lower limit for the integrals in eq. (7) and a further term for the remainder of the angle range. Bull et al. 9) discuss how the accuracy of measuring the exponent y depends on the accuracy of evaluating a. All of these correction factors are associated with the angle differential dO. As the corrected momenta increase the corresponding momentum bin widths should increase by 1/02 . In order to distinguish between fluxes at p =/3 and p = 2/3 one may have to choose a bin width less than p/5 at p =/3, depending on the rate of change of C. The significance of C will then also depend on the concomitant statistics. 4. Corrections to the charge ratio
With the same assumptions as in the previous section the measured charge ratio is
em(~b) = {fom~et(o)'OT-2expl-½(O-dp~21dO + L
\ tr / l
D/a¢ TeV/c.
10 - 6 4 ( n x )
Bx/x
317
MEASUREMENTS
X { ;im°~ 0'-- 2 e x p [ - - ½ ( ~ - - ~ ) 2 ] dO +
i 0+4)
z
-1
Rt(O)'Oe-2expI-~(--T-)]dO },
(9) where the true charge ratio
(Omax+ 1). i0"\
-
That is the correction factor C will markedly deviate from unity when scattering is appreciable. At higher momenta these corrections may still be significant, depending on the value chosen for 0m~x. Baber et al. 8) compare quantitatively sample incident spectra, after correcting for these instrumental
Rt(0 )
F(O+)
F(O_)"
To evaluate this one has to add a further assumption, that of the dependence of R t o n 0. Aurela et al. 1°) have obtained analytic solutions for the integrals by assuming a constant value of R t when y equals 2 and 3 and 0max =oo. This will only be justified when the
318
A. F R O H L I C H
corrected charge ratio is i n d e p e n d e n t o f m o m e n t u m , scattering is negligible a n d the most p r o b a b l e value o f ~ is used. F o r a m o r e accurate calculation one w o u l d have to resort to a m u l t i - d i m e n s i o n a l p a r a m e t e r space '3) a n d use the X2 t e s t to give a best fit at each m o m e n t u m measured. Baber et al. '1) used the same m e t h o d as in ref. 8 to determine the best fit to the total flux. Then they assumed a constant charge ratio and, after allowing for instrumental errors, calculated quantitatively a correction factor to their measured charge ratio. They found that the corrected charge ratio was relatively insensitive to the assumed charge ratio. The correction was only different from unity at m o m e n t a near to the mdm, when in fact their corrected charge ratio a p p e a r e d to rise. H o w e v e r the uncertainty in the resultant charge ratio due to p o o r statistics and instrumental uncertainties was such that the corrected charge ratio could be m o m e n t u m i n d e p e n d e n t in the range examined. It m a y be worth noting that using eq. (9) with the same assumptions as A u r e l a et al. a similar small correction is found. Similar results were r e p o r t e d by Palmer and Nash'2).
5. Conclusions M e t h o d s o f correcting observed spectra are subject to the a d o p t e d flux models, to the effects o f C o u l o m b scatter a n d inaccuracies in spatial resolution. In
principle it would be possible by an iterative m e t h o d to a p p r o a c h the most p r o b a b l e spectra but the goodness o f fit should then be c o m p a t i b l e with the available statistics and within the possible ranges o f the various p a r a m e t e r s used.
References 1) A . W . Wolfendale, Muons, Rapporteur Paper, Proc. l l t h Intern. Conf. Cosmic rays (Budapest, 1969). 2) O. C. Allkofer, L. Carstensen, W. D. Dau, W. Heinrich, E. Kraft and M. Weinert, Nucl. Instr. and Methods 83 (1970) 31"7, table 4. a) p . j . Hayman and A . W . Wolfendale, Proc. Phys. Soc. 80 (1962) 710. 4) R.J. Stefanski, R . K . Adair and H. Kasha, Phys. Rev. Letters 20 (1968) 950.
a) G.H. Lowe, J.H. Parker, R.O. Stenerson and H.E. Bergeson, Proc. l lth Intern. Conf. Cosmic rays (Budapest, 1969) to be published Acta Phys. Hungaria (1971). 6) B. B. Rossi and K. I. Greisen, Rev. Mod. Phys. 13 (1941) 240. 7) G. Charpak, Ann. Rev. Nucl. Sci. 20 (1970) 195. 8) S. R. Baber, W. F. Nash and B. C. Rastin, Nucl. Phys. B4 (1968) 539. 9) R. M. Bull, W. F. Nash and B. C. Rastin, Nuovo Cimento 24 (1962) 1096. 10) A. M. Aurela, P.K. MacKeown and A.W. Wolfendale, Proc. Phys. Soc. 89 (1966) 401. 11) S. R. Baber, W. F. Nash and B. C. Rastin, Nucl. Phys. B4 (1968) 549. 12) N.S. Palmer and W.F. Nash, Proc. 1lth Intern. Conf. Cosmic rays, Paper MU 1-5, (Budapest, 1969) to be published Acta Phys. Hungaria (1971). la) R. J. R. Judge and W. F. Nash, Can. J. Phys. 46 (1968) 907
ON THE PRECISION OF MOMENTUM
MEASUREMENTS WITH A MAGNETIC SPECTROMETER A. FROHLICH
Department o f Physics and Astronomy, TeI-Aviv University, R a m a t Aviv, Tel-Aviv, Israel
Received 11 June 1971 Methods of analysing the data from magnetic spectrometers are examined. The effects of Coulomb scatter counter efficiencies and assumed spectrum exponent on the observed spectra are
considered, and how they might be incorporated into the analytic methods used for correcting the measured muon flux and charge ratio.
1. Introduction
scattering within the magnet iron and that due to the efficiency and spatial resolution o f the particle detectors. In this paper the m o m e n t a are considered to be high enough that the m o m e n t u m loss in the iron is negligible. When the thickness of the iron is much greater than the radiation length, as is usual, the projected multiple scattering o f relativistic muons has a Gaussian distribution with a standard deviation o f 6)
The study of high energy cosmic ray muons is, at present, the only means o f investigating m u o n production models and interactions at energies o f the order o f magnitude of 1 TeV (1012 eV). The principal instruments used in these experiments are magnetic spectrometers, the majority incorporating solid iron magnets1'2). However the accuracy o f measurements with these instruments has not been fully investigated. A quantity called the m a x i m u m detectable m o m e n t u m , mdm, is variously defined and used as a measure o f the upper limit o f the m o m e n t u m resolution, although measurements at higher m o m e n t a have been made and compensated for instrumental errors3). These corrections have been criticised as being insufficient to ensure enough accuracy~). It is precisely where these corrections become important, at m o m e n t a higher than or o f the same order as the mdm, that the collection statistics are p o o r because of the steep decrease of the m u o n flux with m o m e n t u m , and also where effects such as m u o n isotropy m a y appearS). This work was initiated as part o f the design studies for a large m u o n spectrometer that is to be built at "Fel-Aviv, in order to estimate the precision attainable.
11.287 x 10 - 6 a~ =
/
. ,,' x
rad.
(3)
P The spatial resolution o f a spark chamber also has a Gaussian distribution with a standard deviation of 7) o-c. The efficiency o f operation o f these chambers depends on the operating conditions and the number o f tracks. Fig. 1 shows a schematic outline of a simple
V////////////J//////////////////////////////////////////~//J~
2. Sources of error
The experiments consist o f measuring the angular deviation o f a muon within the magnetic volume and using the equations p = k/0,
(1)
k = 3 x 10 -~° B . x .
(2)
,where B is the mean flux in gauss, x the length o f the trajectory within the magnet and p is in TeV/c. The measurement o f the angular deviation 0 is subject to two kinds o f error; that due to C o u l o m b 315
t
I3
Fig. 1. Schematic spectrometer.
316
A. FROHL1CH
spectrometer with four trays of detectors, each tray containing n spark chambers relatively close together. If the efficiency of each gap is 100% the scatter of the measured angular deviations will be Gaussian with a standard deviation a m = 2ad(Dx/n) rad, without scatter, and the overall standard deviation of the measurement will be a = (a~ + am2){ rad.
(4)
In practice the efficiency of spark chambers is less than 100%. A gap can either produce an acceptable spark(s) or not, that is the probability distribution is binomial. As an example the probability that m out of 12 gaps will fire is
12Cm(~)rn(1--q)12-m with ~ the gap efficiency. With a gap efficiency of 98% all twelve gaps will fire together only in about 80% of the events. Consequently the measurement error a., should be weighted accordingly, to all of the possible combinations of acceptable firings. The probability distribution of the angular deviations resulting from a muon of momentum p = + k/O is P(~b) dq~
/3 is often referred to as the maximum detectable momentum, mdm, the Durham group preferring the value 1.48/3. This phrase is rather misleading because meaningful measurements of momenta greater than this value are feasible. It is suggested that an expression of the type "equivalent noise m o m e n t u m " would be semantically better. 3. Corrections to the flux spectrum
Because of the above effects corrections have to be made to the final observed spectrum. It is assumed that the most probable incident flux is N(p)dp=FP-~dp with a constant value of Y in the range Pn,~,, ~ P ~< oo, and that the value of k in eq. (2) is also constant within this momentum range. Using eq. (5) the observed deflection spectrum would be N'(~b) .
. . . exp - ½ Jo x/(2n)a
+
c
,](2n) a
3.5
If the path lengths of measured tracks within the magnetic volume do not vary appreciably, i.e. k in eq. (2) is constant, one can transform this distribution from the angular plane to the momentum plane:
P(p) d p -
\/2n--
/
exp 2.5
where/3 = k/a and fi = k/O. As can be seen this distribution is not Gaussian; the most probable momentum satisfies the equation
?'=3
2.8
2
2+8
,
(P) = 4
assuming scattering to be negligible. This constancy of k is only valid at high momenta and where the aspect ratio of the spectrometer is large. With a large acceptance aperture the variation of k due to variations of the path length x should be taken into account: not only here but also in the final analysis when converting angles to momentum with the aid of eq. (!). For this reason working in the angular plane is preferred.
2.6
1.5 2.4
2.2 1
7=2 1
2
3
P@
Fig. 2. Correction factor to the flux spectrum with scattering negligible.
ON THE PRECISION
OF M O M E N T U M
using a mean value of a. The second term results from the probability of an apparent charge reversal. The correction factor is then C(qS)= N'((9)/F'W-2 within this range of 0. The value of 0m,x is the limit where one or more of the above assumptions are no longer valid; such as an appreciable variation of k, or where scattering effects predominate such that cr varies with 0, or where the magnetic cut-off of the instrument is no longer negligible. Eq. (7) can be easily solved analytically when y equals 2 or 3: For y = 3 C = e r f ( ~b ~ +½{erf(tl)-erf(t2)} \x/2 ~1 +
a
1
d?x/(2n) 2expI-( /-~-~)21- exp[-(tl)2] -
- exp [ - (t2)2],
(8)
when tl = ,,,/21
, and t2
x / 2 \ tr +
"
For 7 = 2 C = ½[erf(tl) + err(t2)]. Hayman and Wolfendale 3) have used eq. (8) with 0ma x = OO. Even if one can ignore the end effects denoted by a finite 0m~x the actual values of C are sensitive to the assumed value of ),. Fig. 2 shows C as a function of p//3 and 7 under these conditions. One would expect a finite 0max when scattering effects become appreciable, that is when as> am o r Pmin
~ 5.64 x
Using for the dimensions of fig.l x = 3 0 0 c m , D = 250 cm, n = 3 and ac =0.5 mm and assuming a spark efficiency of 100% we find pmi,<0.8 TeV/c. In this situation, where cr~trs, using eqs. (1) and (3) we arrive atp/p = a/q5 = (3.76 x 104)/(Bw/x),a constant for a given trajectory. Thus for measurements at low momenta, with a constant path length in the magnetic flux the limits of eq. (8) are t-
532
effects, to their measurements by the )(2 test. This method will, of course, only optimise the measurements to within the parameters of the flux model chosen a priori. This criticism applies to all of the techniques chosen to correct spectra. If, for example, the exponent changes, as the results of the Utah group would indicateS), these corrective techniques, having a fixed 7, might not show this: instead they would give a best fit to the data from a model having a fixed exponent. Any anticipated changes in y would have to be allowed for by using a finite lower limit for the integrals in eq. (7) and a further term for the remainder of the angle range. Bull et al. 9) discuss how the accuracy of measuring the exponent y depends on the accuracy of evaluating a. All of these correction factors are associated with the angle differential dO. As the corrected momenta increase the corresponding momentum bin widths should increase by 1/02 . In order to distinguish between fluxes at p =/3 and p = 2/3 one may have to choose a bin width less than p/5 at p =/3, depending on the rate of change of C. The significance of C will then also depend on the concomitant statistics. 4. Corrections to the charge ratio
With the same assumptions as in the previous section the measured charge ratio is
em(~b) = {fom~et(o)'OT-2expl-½(O-dp~21dO + L
\ tr / l
D/a¢ TeV/c.
10 - 6 4 ( n x )
Bx/x
317
MEASUREMENTS
X { ;im°~ 0'-- 2 e x p [ - - ½ ( ~ - - ~ ) 2 ] dO +
i 0+4)
z
-1
Rt(O)'Oe-2expI-~(--T-)]dO },
(9) where the true charge ratio
(Omax+ 1). i0"\
-
That is the correction factor C will markedly deviate from unity when scattering is appreciable. At higher momenta these corrections may still be significant, depending on the value chosen for 0m~x. Baber et al. 8) compare quantitatively sample incident spectra, after correcting for these instrumental
Rt(0 )
F(O+)
F(O_)"
To evaluate this one has to add a further assumption, that of the dependence of R t o n 0. Aurela et al. 1°) have obtained analytic solutions for the integrals by assuming a constant value of R t when y equals 2 and 3 and 0max =oo. This will only be justified when the
318
A. F R O H L I C H
corrected charge ratio is i n d e p e n d e n t o f m o m e n t u m , scattering is negligible a n d the most p r o b a b l e value o f ~ is used. F o r a m o r e accurate calculation one w o u l d have to resort to a m u l t i - d i m e n s i o n a l p a r a m e t e r space '3) a n d use the X2 t e s t to give a best fit at each m o m e n t u m measured. Baber et al. '1) used the same m e t h o d as in ref. 8 to determine the best fit to the total flux. Then they assumed a constant charge ratio and, after allowing for instrumental errors, calculated quantitatively a correction factor to their measured charge ratio. They found that the corrected charge ratio was relatively insensitive to the assumed charge ratio. The correction was only different from unity at m o m e n t a near to the mdm, when in fact their corrected charge ratio a p p e a r e d to rise. H o w e v e r the uncertainty in the resultant charge ratio due to p o o r statistics and instrumental uncertainties was such that the corrected charge ratio could be m o m e n t u m i n d e p e n d e n t in the range examined. It m a y be worth noting that using eq. (9) with the same assumptions as A u r e l a et al. a similar small correction is found. Similar results were r e p o r t e d by Palmer and Nash'2).
5. Conclusions M e t h o d s o f correcting observed spectra are subject to the a d o p t e d flux models, to the effects o f C o u l o m b scatter a n d inaccuracies in spatial resolution. In
principle it would be possible by an iterative m e t h o d to a p p r o a c h the most p r o b a b l e spectra but the goodness o f fit should then be c o m p a t i b l e with the available statistics and within the possible ranges o f the various p a r a m e t e r s used.
References 1) A . W . Wolfendale, Muons, Rapporteur Paper, Proc. l l t h Intern. Conf. Cosmic rays (Budapest, 1969). 2) O. C. Allkofer, L. Carstensen, W. D. Dau, W. Heinrich, E. Kraft and M. Weinert, Nucl. Instr. and Methods 83 (1970) 31"7, table 4. a) p . j . Hayman and A . W . Wolfendale, Proc. Phys. Soc. 80 (1962) 710. 4) R.J. Stefanski, R . K . Adair and H. Kasha, Phys. Rev. Letters 20 (1968) 950.
a) G.H. Lowe, J.H. Parker, R.O. Stenerson and H.E. Bergeson, Proc. l lth Intern. Conf. Cosmic rays (Budapest, 1969) to be published Acta Phys. Hungaria (1971). 6) B. B. Rossi and K. I. Greisen, Rev. Mod. Phys. 13 (1941) 240. 7) G. Charpak, Ann. Rev. Nucl. Sci. 20 (1970) 195. 8) S. R. Baber, W. F. Nash and B. C. Rastin, Nucl. Phys. B4 (1968) 539. 9) R. M. Bull, W. F. Nash and B. C. Rastin, Nuovo Cimento 24 (1962) 1096. 10) A. M. Aurela, P.K. MacKeown and A.W. Wolfendale, Proc. Phys. Soc. 89 (1966) 401. 11) S. R. Baber, W. F. Nash and B. C. Rastin, Nucl. Phys. B4 (1968) 549. 12) N.S. Palmer and W.F. Nash, Proc. 1lth Intern. Conf. Cosmic rays, Paper MU 1-5, (Budapest, 1969) to be published Acta Phys. Hungaria (1971). la) R. J. R. Judge and W. F. Nash, Can. J. Phys. 46 (1968) 907